Derivative of Real Area Hyperbolic Secant of x over a

From ProofWiki
Jump to navigation Jump to search

Theorem

$\dfrac {\map \d {\map \arsech {\frac x a} } } {\d x} = \dfrac {-a} {x \sqrt{a^2 - x^2} }$

where $0 < x < a$.


Proof

Let $0 < x < a$.

Then $0 < \dfrac x a < 1$ and so:

\(\ds \frac {\map \rd {\map \arsech {\frac x a} } } {\rd x}\) \(=\) \(\ds \frac 1 a \frac {-1} {\frac x a \sqrt {1 - \paren {\frac x a}^2} }\) Derivative of $\arsech$ and Derivative of Function of Constant Multiple
\(\ds \) \(=\) \(\ds \frac 1 a \frac {-a} {x \sqrt {\frac {a^2 - x^2} {a^2} } }\)
\(\ds \) \(=\) \(\ds \frac {-a} {x \sqrt {a^2 - x^2} }\)


$\arsech \dfrac x a$ is not defined when $x \le 0$ or $x \ge a$.

$\blacksquare$


Also see